Here
is a recipe for pipes that is reasonably simple to follow. The completed
pipe has two parts, the resonator body with cut-up and foot, and a screwed-on
cover forming the outside flue edge, ears and 'intonation roll', see the
attached drawing. Highlights in the design are the ease to assemble it
and to control the air band thickness. Most of the voicing work is concentrated
to the readily unscrewed cover piece.

In classical pipe making you determine the various
measures of a pipe by proportional scaling in a graphical diagram. These
diagrams are based on experience from historic pipes and are often jealously
guarded trade secrets of the pipe makers.

The approach here is instead to give simple formulas.
You have to make a small number of basic selections for pipe type, width
scale, note number, and for practical things like foot design.

The rational use of the formulas is to enter them
in a computer spreadsheet program and then you can easily develop lists
for all parts dimensions for a complete pipe rank. Click here to view or
download the Microsoft Excel spreadsheet file, violpip.xls.

Key dimensions

Some key dimensions are found from elementary physical relations where
the value for a particular note is selected in terms of its semitone number
n on the equally tempered scale. The assignment here is MIDI oriented such
that note number n=69 is for the standard a'=440 Hz. Then the 'middle c',
or 'keyhole c', is number 60.

In classical organ terminology you talk about pipe ranks being 16 ft,
8 ft, etc. Basically this is oriented toward the organist rather than the
pipe maker. That measure refers to the size of an open pipe (that possibly
is) activated by the C key on the keyboard (or pedal). On a piano the standard
a' is always the one next right of the 'middle c'. But in an organ this
is the case only for an 8 ft stop. Other size ranks produce notes transposed
one or several octaves, or even fractions of octaves. The 8 ft C=65.41
Hz (or 16 ft c, 4 ft 'C) corresponds to our note number n=36.

Other key dimensions are defined from the designer's selections of scale
for a rank, and from some practical considerations.

L = nominal Length = half wavelength

Depends only on which note to render. The frequency of note number
n will be f(n)=440*2^((n-69)/12) and the half wavelength will be L(n)=c/2f(n)
where c is the speed of sound 343.32 m/s @ 20 deg. C (increasing 0.59 m/s
per deg. C - tuning changes 3 ct per deg. C). If we combine the formulas
and insert c we have

L(n) = 0.3901 * 2^((69-n)/12) [meters]

L(n) = 15.36 * 2^((69-n)/12) [inches

This tells how long the resonator should be in theory for an open pipe.
To arrive at the practical length you have to account for several things;
this is developed in the following section.

W = inside Width/diameter, andM = halving number

A major determinant of tonal timbre, designers choice. Suggested 'normal'
value for standard 440 Hz a is W(69) = L(69)/12 = 0.0325 m = 1.28
in. Larger values gives pipes with a loud fundamental and weak higher harmonics
in relative terms. Smaller values give weaker pipes rich in harmonics,
for instance a violin type pipe may have W(69) = L(69)/18 = 0.022 m = 0.87
in. For other notes you have to select a 'halving number' M, which means
the number of semitones you have to go up to have a pipe with W(n+M) =
W(n)/2. For any pipe then W(n) = W(69) * 2^((69-n)/M). Conventional practice
is M = 16 to 24, but also M = infinity (meaning all pipes in a rank have
the same diameter as W(69) ) has been used.

T = tube wall Thickness

Designers choice, essentially to fulfill esthetical and practical requirements
like stability and ease of building. Rule of thumb is around T = W/6, perhaps
more with small pipes (W < 12 mm, 0.5 in), less with large ones (W >
50 mm, 2 in).

F = Foot mount height

Designers choice, to suit appearance and method of connection to the
wind chest. May be the same for all pipes in a rank. The drawing shows
a somewhat unconventional example with a conically reamed hole in the pipe
foot and a matching tapered metal tube which is integral with the wind
chest.

H = cut-up HeightD = flue air band thickness

These parameters are essential for the proper function and timbral
characteristics of the pipe. They are related to frequency, power, and
blowing pressure. Refer to the Ising intonation number and associated dimensioning
chart at http://foxtail.com/Tech/isint.html.
Here it is difficult to give any specific recommendations to follow, there
are many alternative ways to specify. One simple rule that I have used
is to put D=W/100 and then proceed with the Ising formula to find H given
frequency and pressure. Another possible rule is to specify that (frequency)*(input
power) = constant, having in mind that the input power is proportional
to the flue area D*W. Incidentally the first rule gives this result when
M=24.

E = tuning cut down depth

Applies to the open end of open pipes only. See below.

Length of a pipe

There are some factors you must account for when calculating the physical
length. First thing is to decide whether to make an open pipe or a closed
pipe. Generally open pipes are somewhat louder and have stronger harmonics,
richer timbre, than closed pipes.

The length of an open pipe is somewhere near L, at its basic resonance
it contains a half wave of the sound. To fine tune it a convenient method
is to cut down one wall of the pipe at its open end and to partly cover
that opening with a plate that you can slide up or down.

The length of a closed pipe is somewhere near L/2, at its basic resonance
it contains a quarter wave of the sound. To fine tune it you plug its end
with an adjustable stopper. The pipe must have some extra length (W suggested)
to hold the length of this stopper.

Another factor is the so called 'end correction'; a real pipe will resonate
at a lower frequency than one might believe from its physical length because
some air outside the ends of the pipe takes part in the resonance motion.
Already 150 years ago Cavaillé-Coll gave a good rule: with an open,
square pipe you should subtract twice the width from the nominal length.
With a closed pipe you subtract only once the width. The latter rule is
not accurate with an extreme width where we tend to get a Helmholtz cavity
resonator (like the ocarina and many whistles) rather than a transmission
line tube resonator.

For tuning the pipe a reasonable tuning range is +/-50 ct, a half semitone
up or down. Then you have to make the untuned pipe to be 50 ct low, that
is, you should increase L(n) by 3%.

Finally, adding W+F for the flue and foot areas we arrive at the total
length of open and closed pipes to be respectively

Lop(n) = 1.03L(n) -2W(n)+W(n)+F
= 1.03L(n)-W(n)+F,

Lcl(n) = 1.03L(n)/2- W(n)+W(n)+F+W(n) = 0.52L(n)+W(n)+F.

In practical tuning measurements I found that opening up one side
by the length E is approximately equivalent to shortening the pipe by dL=E*E/(E+W).
When the opening E is less than the width W this has a comparatively small
influence, but when larger than W the influence is about proportional.
To tune up to 50 ct high you should decrease L(n) by 3%, a total difference
of 6% from the previously incremented L(n). But since cutting only one
wall is less efficient, solving the empirical equation above gives the
open pipe tuning cut down E = 0.03L*(1+sqrt(1+4W/L)). This ranges from
E=0.06L for an extremely narrow pipe to 0.072L for a very wide one having
L/W=4. Forget about this small variation and just select E=0.07L.

Be warned that varying the cut-up height H and the possible insertion
of an intonation roll (f) will change the tuning of the pipe. Also fabrication
tolerances give their share. The final decision of total length should
be left to the last moment when you voice the pipe. The resonator length
measures given appear to be conservative such that you can expect the final
length to be a little shorter than indicated.

Parts dimensions

The following table lists the dimensions of all parts, expressed in
the key dimensions. See Figure 1 below for the parts referenced by "RefDes"
(reference designation). The indication "+x" means you should add a small
extra margin (1..3 mm, 1/16..1/8 in) which is to be shaved off later, during
assembly and finishing. The lengths for the body pieces (a) and (b) have
two alternatives, select the appropriate one for closed pipes or for open
pipes respectively.

RefDes Qty Length
Width Thickness

Closed: front, back: a 2
0.52L+F+W+x W+2T+x T

sides: b 2 0.52L+F+W+x
W+x T

Open: front, back: a
2 1.03L+F-W+x W+2T+x T

sides: b 2 1.03L+F-W+x
W+x T

Bottom plug
c 1 F+W+x
W W+x

Cover
d 1 F+W+x
W W/2+x

Ears
e 2 F+2W+x
W/2+x T

Roll
f 1 W
W/4 W/4

Gasket
g 1 F+2W+x
W+2T+x D

Closed: Stopper plug h 1
0.7W W-
W-

Open: Tuning plate i
1 0.07L+3T W+T/2

Tuning cut down E
0.07L

Spreadsheet

An example framework spreadsheet for MS Excel including the key dimension
formulas is violpip.xls . Here you can
enter your specific selections and also decide whether to use meters or
inches.

Assembly

To insure alignment it is wise to drill the axial foot hole in the bottom
plug (c) before assembly, perhaps even in a lathe.

Clamp cover (d) between upper ends of sides (b) in order to keep distance.
Glue plug (c) between the sides (b) and clamp. Be very careful to put all
pieces in parallel, align by simultaneous pressing parts against a flat
support to avoid any warp. Let dry and then plane the front and back surfaces
to be accurately flat and to measure.

Keep cover (d) clamped as a distance holder between the ends of the
sides (b). Glue on the front and back pieces (a), clamp and let dry. Remove
(d) and shave off the surplus width of (a) with a plane to make the tube
sides flat.

Do the outside beveling and inside wedge wise cut on the cover (d).
Try to make the flue edge precisely aligned to the cover inside surface.
Glue on ears (e) and clamp, align against a flat support. When dry, shave
off the surplus to make both sides flat and thickness to measure. Here
is the time to fine adjust the inside wedge cut such that the outer
flue edge is precisely flat with the inside of the cover.

Drill the hole connecting the outside to the foot hole. The drawing
shows the front piece to be a wide one, covering the whole front. Of course
you may instead elect to let one of the pieces (b) form the front. If you
do so, then the natural thing would be to have made this front piece shorter
than the other pieces (a) and (b) by the tuning cut down measure E.

Draw accurately the location of the cut-up on the outside of the tube,
use a stick to check against the top of the internal bottom plug. Drill
through the wall one or several places and shape the rectangular opening
with a knife and a file. Then sharpen the labium edge by oblique cutting
away on the inside. Most easily done with a very sharp, pointed knife.
Pull the knife while cutting small chips at a time. Finish with a small
file. This is the most difficult part of the making and it is wise to protect
the flue edge against dents by clamping a thin metal plate over the foot
area. There is no merit to make the labium extremely sharp, you can leave
an edge width of perhaps 0.5 mm or 1/64 inch.

Now cover the cut-up with adhesive tape and fill the upright pipe with
thin hot hide glue (or whatever sealing agent you may prefer), leave for
perhaps 30 seconds and then pour it out again. Remove the tape, let the
excess sealant drip off and let the pipe dry in upside down position.

The gasket defines the air band thickness D. It can be made from different
materials, for instance cardboard, leather, or veneer. Let it have some
oversize thickness, cut it out with some extra margin and glue it to the
cover. When dry, sand the outsides on a flat sandpaper on the table to
be flush with the cover ears. Inside the ears, trim with a sandpaper file
or a knife.

Mark the places for screw holes on the cover and clamp it in place on
the pipe body. Drill the screw holes through the cover into the body with
a drill corresponding to the core size of your screws. Then remove the
cover and re-drill it to match the screw outer diameter.

For an open pipe, cut the narrow grooves for the tuning slider plate
using a hacksaw blade. The depth of these grooves may be some T/3 and they
should extend 2E down the tube. Draw a line across the front where the
tuning cut down should end, distance E from top. Drill a sequence of holes
just above this line and saw along the inner sides. Break loose the chip
and finish the edges of the cut down with knife and file. Cut the slider
plate from 0.5..1 mm sheet metal of your choice. If you cut it with shears
you may have to flatten the plate afterwards, use a wooden mallet to avoid
hammer marks. File notches at its upper end corners and fold the upper
edge into a handle. Finish the sides of the plate with a file and match
it to its grooves in the pipe. Most probably you will have to go a second
round with the hacksaw in the lower part of the grooves. Bend the plate
slightly concave towards the front such that it slides with appropriate
friction.

For a closed pipe, make a stopper plug to match the interior of the
pipe when clad with a soft leather strip around its circumference. Make
a handle to taste. You should bevel all sides of the stopper, both at top
and bottom, such that its side and front views about fit in a circle. This
in order to avoid breaking the pipe in case you some later time have to
rock the stopper loose because it is hard stuck. Glue on the leather strip
around the stopper and be careful to apply glue only as a narrow string
along the extreme low end of the stopper. To fine adjust how stiff the
stopper runs inside the pipe you slide paper strips into the unglued area
between the plug and the leather.

Voicing

For voicing the primary parameter is the thickness of the gasket (g).
This is easily controlled sliding the cover over a sandpaper, flat on the
table. If you install a 'harmonic brake' in form of a roll (f) you can
use a relatively high intonation number (big D, small H, high pressure)
without over blowing the pipe. Then the precise position of the roll is
critical and has to be found by trial and error. When satisfied you can
lock the roll in place with drops of cyanoacrylate glue and/or needles
drilled in through the ears (e).

The result

Three pipes are shown in the photograph (top of page); from left a'=440,
a''=880, and a'''=1760 Hz. All are of fairly narrow scale, they have W(69)=L(69)/18,
M=24, T=4 mm, F=20 mm. The cover of the middle one is removed to show the
inside. (The fixing screw placement on the left pipe is not recommended).

The left one has H=12 mm, D=0.4 mm and when open at its far end it sounds
like o440.wav when blowing pressure
is 2 kPa (8 inches WC). Operating it as a closed pipe for the same note
(special stopper with a long handle pushed about half way down the resonator)
changes its sound into c440.wav . A bigger one in
the series sounds like o220.wav , and the
two smaller ones in the picture like o880.wav
and o1760.wav .